It is the nuclear spin of the 13C isotope that is exploited in the NMR experiments on fullerenes.

Although small in abundance, the 13C isotope occurs on approximately half of the C60 molecules synthesized using the natural abundance of carbon isotopes as shown in Table 4.25. The probability pm for m isotopic substitutions to occur on an nc atom fullerene C„c is given by where x is the fractional abundance of the isotope and the binomial coefficient appearing in Eq. (4.8) is given by

For the natural abundance of carbon isotopes (x = 0.01108), we obtain the results for the C60 and C70 molecules listed in Table 4.25 and shown graphically in Fig. 4.2. Less than 1% of the C^q and C70 fullerenes have more than three 13C isotopes per fullerene. Also included in the table are the corresponding results for an isotopic enrichment to 5% and 10% 13C. The results in Table 4.25 show that as x increases (and also as nc increases in C„c), the peak in the distribution moves to larger m values and the distribution gets broader. These conclusions follow from binomial statistics, where the average in and the standard deviation Am = ((m — m)2)l/2 give m — ncx (4.10)

respectively. For example, for a fullerene with x = 0.05, Eqs. (4.10) and (4.11) yield m = 3 and Am = 1.7. Thus as x increases, so do m and Am, thereby accounting for the broader distribution with increasing x. In general, for isotopically enriched samples, the distribution pm(C„c) is sufficiently shifted and broadened so that graphical displays are desirable, such as shown in Fig. 4.2.

The results of Table 4.25 have important consequences for both symmetry considerations and the rotational levels of fullerene molecules [4.16], and these topics are further discussed in §11.3.3. The molecules containing one or more 13C atoms show much lower symmetry than that of the full Ih point group. In fact, the singly 13C substituted molecule 13CI12C59 has only one symmetry operation, a single reflection plane (point group Clh, see Table 4.26); two or more substitutions generally show no symmetry,

Table 4.25

The probabilities of pm{C„c) for 13C among Qo and C70 fullerenes."

Table 4.25

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